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A282749 Triangle read by rows: T(n,k) is the number of partitions of n into k parts x_1, x_2, ..., x_k such that gcd(x_i, x_j) = 1 for all i != j (where 1<=k<=n).

%I A282749 #18 Jan 05 2025 19:51:41
%S A282749 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,3,1,2,1,1,1,1,2,3,1,2,1,
%T A282749 1,1,1,3,2,3,1,2,1,1,1,1,2,4,2,3,1,2,1,1,1,1,5,2,4,2,3,1,2,1,1,1,1,2,
%U A282749 7,2,4,2,3,1,2,1,1,1,1,6,2,7,2,4,2,3,1,2,1,1,1
%N A282749 Triangle read by rows: T(n,k) is the number of partitions of n into k parts x_1, x_2, ..., x_k such that gcd(x_i, x_j) = 1 for all i != j (where 1<=k<=n).
%C A282749 Column 2 is A023022. It appears that each row ends with some tail portion of the sequence (..., 89, 21, 89, 18, 68, 19, 53, 12, 58, 10, 40, 12, 30, 8, 31, 7, 20, 7, 17, 4, 16, 4, 9, 4, 8, 2, 7, 2, 4, 2, 3, 1, 2, 1, 1, 1). - _Lars Blomberg_ Mar 08 2017
%H A282749 Alois P. Heinz, <a href="/A282749/b282749.txt">Rows n = 1..200, flattened</a> (first 100 rows from Lars Blomberg)
%H A282749 Temba Shonhiwa, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/44-4/quarttemba04_2006.pdf">Compositions with pairwise relatively prime summands within a restricted setting</a>, Fibonacci Quart. 44 (2006), no. 4, 316-323.
%F A282749 It seems that no general formula or recurrence is known.
%e A282749 Triangle begins:
%e A282749 1,
%e A282749 1, 1,
%e A282749 1, 1, 1,
%e A282749 1, 1, 1, 1,
%e A282749 1, 2, 1, 1, 1,
%e A282749 1, 1, 2, 1, 1, 1,
%e A282749 1, 3, 1, 2, 1, 1, 1,
%e A282749 1, 2, 3, 1, 2, 1, 1, 1,
%e A282749 1, 3, 2, 3, 1, 2, 1, 1, 1,
%e A282749 1, 2, 4, 2, 3, 1, 2, 1, 1, 1,
%e A282749 1, 5, 2, 4, 2, 3, 1, 2, 1, 1, 1,
%e A282749 1, 2, 7, 2, 4, 2, 3, 1, 2, 1, 1, 1,
%e A282749 1, 6, 2, 7, 2, 4, 2, 3, 1, 2, 1, 1, 1,
%e A282749 ...
%Y A282749 Cf. A051424 (row sums), A282749 (analog for compositions).
%K A282749 nonn,tabl
%O A282749 1,12
%A A282749 _N. J. A. Sloane_, Mar 05 2017